I was reading the proof of Liouville's theorem for harmonic functions (in $\mathbb{R}^n$) in Wikipedia, but I could not understand where do they use in that proof the assumption that $f$ is bounded.
The proof -
Taken from - https://en.m.wikipedia.org/wiki/Harmonic_function
On
The boundeness of $f$ is used in the proof of the fact that, given two points $x$ and $y$, the average value of $f$ on a large disk centered at $x$ and the average value of $f$ on a large disk (with the same radius) centered at $y$ will go to the same value as the radius goes to $\infty$. When that happens, the symmetric difference of the two discs gets smaller and smaller in proportion to their intersection. Since $f$ is bounded, the average of the function on one disc is then essentially the average of the function on their intersection. Therefore, as the radius goes to infinity, the average over either disc goes to the same number.
Jose's answer gives details of Nelson's original reasoning, but they aren't quite the same as the details in the proof on Wikipedia.
The crucial point in the latter is that we assume, without loss of generality, that $f$ is a nonnegative function (we can assume this because we assumed $f$ is bounded from above or below). Then nonnegativity is used in the first displayed inequality to say that an integral over $B_r(y)$ must be at least as large as the integral over $B_R(x)$, since the latter is a subset of the former.